Brownian Motion on Treebolic Space:Positive Harmonic Functions Tome 66, No 4 (2016), P

R AN IE N R A U L

I ’ I

N S ANNALES

DE L’INSTITUT FOURIER

Alexander BENDIKOV, Laurent SALOFF-COSTE, Maura SALVATORI & Wolfgang WOESS Brownian motion on treebolic space:positive harmonic functions Tome 66, no 4 (2016), p. 1691-1731.

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BROWNIAN MOTION ON TREEBOLIC SPACE: POSITIVE HARMONIC FUNCTIONS

by Alexander BENDIKOV, Laurent SALOFF-COSTE, Maura SALVATORI & Wolfgang WOESS (*)

Abstract. — This paper studies potential theory on treebolic space, that is, the horocyclic product of a regular tree and hyperbolic upper half plane. Relying on the analysis on strip complexes developed by the authors, a family of Laplacians with “vertical drift” parameters is considered. We investigate the positive harmonic functions associated with those Laplacians. Résumé. — Ce travail est dedié à une étude de la théorie du potentiel sur l’espace arbolique, i.e., le produit horcyclique d’un ârbre régulier avec le demi- plan hyperbolique supérieur. En se basant sur l’analyse sur les complexes à bandes Riemanniennes développée par les auteurs, on considère une famille de Laplaciens avec deux paramètres concernant la dérive verticale. On examine les fonctions harmoniques associées à ces Laplaciens.

1. Introduction

We recall the basic description of treebolic space. For many further details on the geometry, metric structure and isometry group, the reader is referred to [3].

We consider the homogeneous tree T = Tp , drawn in such a way that each vertex v has one predecessor v− and p successors. We consider T as a metric graph, where each edge is a copy of the unit interval [0 , 1]. The discrete, integer-valued graph metric dT on the vertex set (0-skeleton) V (T) of T has an obvious “linear” extension to the entire metric graph. We can

Keywords: Tree, hyperbolic plane, horocyclic product, quantum complex, Laplacian, positive harmonic functions. Math. classiﬁcation: 31C05, 60J50,53C23, 05C05. (*) Partially supported by Austrian Science Fund (FWF) projects P24028 and W1230 and Polish NCN grant DEC-2012/05/B/ST 1/00613. 1692 A. Bendikov, L. Saloﬀ-Coste, M. Salvatori & W. Woess

......

Figure 1. A compact piece of treebolic space, with p = 2.(1)

partition the vertex set into countably many sets Hk , k ∈ Z, such that each Hk is countably inﬁnite, and every vertex v ∈ Hk has its predecessor − − v in Hk−1 and its successors in Hk+1. We write [v , v] for the metric edge between those two vertices, parametrised by the unit interval. For − any w ∈ [v , v] we let t = k − dT(w, v) and set h(w) = t. In particular, h(v) = k for v ∈ Hk . In general, we set Ht = {w ∈ T : h(w) = t}. These sets are the horocycles. We choose a root vertex o ∈ H0 . Second, we consider hyperbolic upper half space H, and draw the hori- k zontal lines Lk = {x + i q : x ∈ R}, thereby subdividing H into the strips k−1 k Sk = {x + i y : x ∈ R , q 6 y 6 q }, where k ∈ Z. We sometimes write H(q) for the resulting strip complex, which is of course hyperbolic plane in every geometric aspect. The reader is invited to have a look at the respective ﬁgures of Tp and “sliced” hyperbolic plane H(q) in [3]. Treebolic space with parameters q and p is then

(1.1) HT(q, p) = {z = (z, w) ∈ H × Tp : h(w) = logq(Im z)} .

Thus, in treebolic space HT(q, p), inﬁnitely many copies of the strips Sk are glued together as follows: to each vertex v of T there corresponds the bifurcation line

k Lv = {(x + i q , v): x ∈ R} = Lk × {v} , where h(v) = k .

(1) This ﬁgure also appears in [2].

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Along this line, the strips

k−1 k − Sv = (x + i y, w): x ∈ R , y ∈ [q , q ] , w ∈ [v , v] , h(w) = logq y − and Su , where u ∈ V (T) with u = v, are glued together. The origin of our space is o = (0, o). In [3], we outlined several viewpoints why this space is interesting. It can be considered as a concrete example of a Riemannian complex in the spirit of Eells and Fuglede [15]. Treebolic space is a strip complex (“quantum complex”) in the sense of [2], where a careful study of Laplace operators on such spaces is undertaken, taking into account the serious subtleties arising from the singularites of such a complex along its bifurcation manifolds. In our case, the latter are the lines Lv , v ∈ V (T). The other interesting feature is that treebolic spaces are horocyclic products of H(q) and Tp , with a structure that shares many features with the Diestel-Leader graphs DL(p, q) [13] as well as with the manifold (Lie group) Sol(p, q), the horocyclic product of two hyperbolic planes with curvatures −p2 and −q2, respectively, where p, q > 0. Compare with Woess [25], Brofferio and Woess [7], [8] and with Brofferio, Salvatori and Woess [6]; see also the survey by Woess [26]. The graph DL(p, p) is a Cayley graph of the Lamplighter group Zp oZ. Similarly, the Baumslag-Solitar group BS(p) = ha, b | ab = bpai is a prominent group that acts isometrically and with compact quotient on HT(p, p). In the present paper, we take up the thread from [3], where we have investigated the details of the metric structure, geometry and isometries before describing the spatial asymptotic behaviour of Brownian motion on treebolic space. Brownian motion is induced by a Laplace operator ∆α,β , where the parameter α is the coefficient of a vertical drift term in the interior of each strip, while β is responsible for (again vertical) Kirchhoff type bifurcation conditions along the lines Lv . The serious task of rigorously constructing ∆α,β as an essentially self-adjoint diffusion operator was undertaken in the general setting of strip complexes in [2]. For the specific case of HT, it is explained in detail in [3]. We shall recall only its basic features, while we shall use freely the geometric details from [3]. Here, we face the rather difficult issue to describe, resp. determine the positive harmonic functions on HT(q, p). In §2, we recall the basic features of treebolic space and the family of Laplacians with Kirchhoff conditions at the bifurcation lines. In §3, we start with a Poisson representation on “rectangular” sets which are compact. Then we obtain such a representation on simply connected sets which are unions of strips and a solution of the Dirichlet problem on those sets. (For the technical details of the main

TOME 66 (2016), FASCICULE 4 1694 A. Bendikov, L. Saloﬀ-Coste, M. Salvatori & W. Woess results presented here, the reader is referred to the respective sections.) For the following, let Ω be the union of all closed strips adjactent to Lo and Ω Ω its interior, and let µw be the exit distribution from Ω of Brownian motion starting at w ∈ Ω. Theorems 3.9& 3.11. (I) Let h be continuous on Ω and harmonic on Ω, and suppose that on that set, h grows at most exponentially with respect to the metric of HT. Then for every w ∈ Ω, Z Ω h(w) = h(z) dµw(z) . ∂Ω (II) Let f be continuous on ∂Ω, and suppose that on that set, f grows at most exponentially with respect to the metric of HT. Then Z Ω h(w) = f(z) dµw(z) , w ∈ Ω ∂Ω deﬁnes an extension of f that is harmonic on Ω and continuous on Ω.

The diﬃculties arise from the singularities at the bifurcation lines plus the fact that Ω is unbounded in the horizontal direction. We obtain that Ω the law µ = µw of the random walk induced by Brownian motion on the union LT of all bifurcation lines has a continuous density and exponential tails. The main result of §4 is as follows.

Theorem 4.7.— Restricting positive harmonic functions on HT to LT yields a one-to-one relation with the positive harmonic functions of the random walk driven by µ on LT.

This is of interest, because the disconnected subspace LT ⊂ HT and that random walk are invariant under the transitive action of the isometry group of HT. In §5, we use Martin boundary theory to prove the following decomposition.

Theorem 5.2.— Every positive harmonic function h on HT has the form h(z) = hH(z) + hT(w) , z = (z, w) ∈ HT , where hH is non-negative harmonic on H(q) and hT is non-negative harmonic on Tp. Given the drift parameters α and β, there is a formula for the vertical drift `(α, β) ∈ R, and |`(α, β)| is the rate of escape of our Brownian motion.

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Theorem 5.6.— Suppose that p > 2. Then the Laplacian ∆α,β on HT has the weak Liouville property (all bounded harmonic functions are constant) if and only if `(α, β) = 0. Finally, in the situation when the projection of the Laplacian on the hyperbolic plane is smooth, we can describe all minimal harmonic functions explicitly. The techniques that we employ here can be applied to obtain similar results on other types of strip complexes. The simplest one is where one replaces “sliced” hyperbolic by Euclidean plane. Very close to the present study is the case where one takes hyperbolic upper half space Hd in arbitrary dimension, subdivided by level sets (horospheres) of the Busemann function with respect to the boundary point ∞. Also of interest: to replace the tree by the 1-skeleton of a higher-dimensional Aed-building, or just to take diﬀerent types of level functions on a regular tree; compare e.g. with the space considered by Cuno and Sava-Huss [12].

2. Laplacians on treebolic space

We brieﬂy recapitulate the most important features of HT, in particular, the construction of our family of Laplace operators ∆α,β on HT(q, p) with parameters α ∈ R and β > 0. For more details, see [3]. The rigorous construction is carried out in [2]. For any function f : HT → R, we write fv for its restriction to Sv . For − (z, w) ∈ Sv , the element w ∈ [v , v] is uniquely determined, so that we can omit w and write fv(z) = f(z, w): formally, fv is deﬁned on Sk ⊂ H, where k = h(v).

Definition 2.1. — We let C∞(HT) be the space of those continuous functions f on HT such that, for each v ∈ V (T), the restriction fv on Sv m n o has continous derivatives ∂x ∂y fv(z) of all orders in the interior Sv which satisfy, for all R > 0, m n h(v)−1 h(v) sup |∂x ∂y fv(z)| : |Re z| 6 R, q

Thus, on each strip Sv , each partial derivative has a continuous extension m n ∂x ∂y fv(z) to the strip’s boundary. However, except for m = n = 0, when − m n m n w = v, we do in general not have that ∂x ∂y fw = ∂x ∂y fv on Lv = Sv∩Sw . The hyperbolic gradient 2 2 ∇fv(z) = y ∂xfv(z) , y ∂yfv(z)

TOME 66 (2016), FASCICULE 4 1696 A. Bendikov, L. Saloﬀ-Coste, M. Salvatori & W. Woess

is deﬁned in the interior of each strip. On any bifurcation line Lv , we have to distinguish between all the one-sided limits of the gradient, obtaining the family − ∇fv(z) and ∇fw(z) for all w ∈ V (T) with w = v , (z, v) ∈ Lv . ∞ For any open domain Θ ⊂ HT, we let let Cc (Θ) be the space of all functions in C∞(HT) which have compact support contained in Θ . We let [ o [ o (2.1) LT = Lv and HT = Sv = HT \ LT . v∈V (T) v∈V (T) The area element of HT is dz = y−2dx dy for z = (z, w) ∈ HTo , where z = x + i y and dx, dy are Lebesgue measure: this is (a copy of) the hyperbolic upper half plane area element. The area of the lines Lv is 0. For α ∈ R , β > 0, we deﬁne the measure mα,β on HT by

dmα,β(z) = φα,β(z) dz , with (2.2) h(v) α φα,β(z) = β y for z = (x + i y, w) ∈ Sv \ Lv− , − where v ∈ V (T), that is, w ∈ (v , v] and logq y = h(w). Definition 2.2. — For f ∈ C∞(HT) and z = (x + i y, w) ∈ HTo , 2 2 2 ∆α,βf(z) = y (∂x + ∂y )f(z) + α y ∂yf(z) . ∞ ∞ Let Dα,β,c be the space of all functions f ∈ Cc (HT) with the following properties. k o (i) For any k, the k-th iterate ∆α,βf, originally deﬁned on HT , admits ∞ a continuous extension to all of HT (which then belongs to Cc (HT) k and is also denoted ∆α,βf). k (ii) The function f, as well as each of its iterates ∆α,βf, satisﬁes the bifurcation conditions X (2.3) ∂yfv = β ∂yfw on Lv for each v ∈ V (T) . w : w−=v ∞ Proposition 2.3 ([2]).— The space Dα,β,c is dense in the Hilbert 2 ∞ space L (HT, mα,β). The operator (∆α,β, Dα,β,c) is essentially self-adjoint 2 in L (HT, mα,β). We also write ∆α,β, Dom(∆α,β) for its unique self-adjoint extension. Basic properties of this Laplacian and the associated heat semigroup are derived in [2]. In particular, there is the positive, continuous, symmetric heat kernel hα,β(t, w, z) on (0, ∞) × HT × HT such that for all f ∈ Cc(HT), Z t∆α,β e f(z) = hα,β(t, w, z) f(z) dmα,β(z) . HT

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∆α,β is the inﬁnitesimal generator of our Brownian motion (Xt)t>0 on HT. It has inﬁnite life time and continuous sample paths. For every starting α,β point w ∈ HT, its distribution Pw on Ω = C([0, ∞] → HT) is determined by the one-dimensional distribution Z Z α,β Pw [Xt ∈ U] = hα,β(t, w, z) dmα,β(z) = pα,β(t, w, z) dz , U U where U is any Borel subset of HT and

(2.4) pα,β(t, w, z) = hα,β(t, w, z) φα,β(z) with φα,β as in (2.2). We have the projections πH : HT → H , πT : HT → T and πR : HT → R , where for z = (z, w),

H T R π (z) = z , π (z, w) = w , and π (z) = logqIm(z) = h(w) . On several occasions it will be useful to write Re z =Re z for z = (z, w) ∈ HT .

The “sliced” hyperbolic H(q) plane is the treebolic space HT(q, 1): the H tree is the bi-inﬁnite line graph. In particular, we have the operator ∆α,β T on H. We also have a Laplacian ∆α,β on the metric tree, whose rigourous construction (in the same way as above) is considerably simpler. Note the diﬀerent parametrisation in T, where each edge [v−, v] corresponds to the real interval [h(v) − 1 , h(v)]. We write fv for the restriction of f : T → R to [v−, v], which depends on one real variable. The analogue of the measure of (2.2) is

T T dmα,β(w) = φα,β(w) dw , with (2.5) T h(v) (α−1)h(w) − φα,β(w) = β q log q for w ∈ (v , v] , where v ∈ V (T) and dw is the standard Lebesgue measure in each edge. The space C∞(T) is as in Deﬁnition 2.1, with the edges of T in the place of the strips. Deﬁnition 2.2 and the bifurcation condition (2.3) are replaced T ∞ by the following: every f ∈ Dom(∆α,β) ∩ C (T) must satisfy for every v ∈ V (T) 0 X 0 fv(v) = β fw(v) and − (2.6) w : w =v 1 α − 1 ∆T f = f 00 + f 0 in the open edge (v−, v) . α,β (log q)2 log q We also have this on the real line, by identifying R with the metric tree with vertex set Z and degree 2. The edges are the intervals [k−1 , k], k ∈ Z,

TOME 66 (2016), FASCICULE 4 1698 A. Bendikov, L. Saloﬀ-Coste, M. Salvatori & W. Woess

R and the Laplacian is ∆α,β . Its deﬁnition as a diﬀerential operator in each open interval (k − 1 , k) is as in (2.6), and the bifurcation condition now is f 0(k−) = β f 0(k+) for all k ∈ Z. We have the following.

Proposition 2.4. — Let (Xt) be Brownian motion on HT(q, p) with infinitesimal generator ∆α,β . H H (a) Zt = π (Xt) is the Markov process on H(q) with generator ∆α,βp. T T (b) Wt = π (Xt) is the Markov process on Tp with generator ∆α,β. R R (c) Yt = π (Xt) is the Markov process on on R with generator ∆α,βp. We also need to recall further features of the geometry of HT, as well as 0 0 of T and H. For two points w, w ∈ T, their confluent w f w is the unique element v on the geodesic path w w0 where h(v) is minimal. In the specific case when w and w0 lie on a geodesic ray spanned by a sequence of vertices that are successive predecessors, the confluent is is one of w and w0, while otherwise it is always a vertex. Analogously, for two points z, z0 ∈ H, we let z ∧ z0 be the point on the hyperbolic geodesic z z0 where the imaginary part is maximal. Recall that hyperbolic geodesics lie on semi-circles which are orthogonal to the bottom boundary line R, resp. on vertical straight lines. It is a straightforward (Euclidean) exercise to see that 0 0 0 (2.7) |Re z −Re z | 6 2 Im z ∧ z , z, z ∈ H . Returning to the tree, the boundary ∂T of T is its space of ends. Each geodesic ray in T gives rise to an end at infinity, and two rays have the same end if they coincide except for initial pieces of finite length. In our view on T as in [3, Figure 3], the tree has one end $ at the bottom, and all other ends at the top of the picture, forming the boundary part ∂∗T. For any w ∈ T and ξ ∈ ∂T, there is a unique geodesic ray w ξ starting at w and having ξ as its end. Analogously, for any two distinct ξ, η ∈ ∂T, there is a unique bi-infinite geodesic η ξ such that if we split it at any point, one part is a ray going to ξ and the other a ray going to η. If both of them belong to ∗ ∂ T then we can also define ξ f η as the element v on the geodesic where h(v) is minimal; it is a vertex. If ξ ∈ ∂∗T then the “vertical” geodesic $ ξ is the side view of a copy of the hyperbolic plane sitting in HT, namely Hξ = {(z, w) ∈ HT : w ∈ $ ξ}. The metric dHT of HT is induced by the hyperbolic arc length inside each strip: let z1 = (z1, w1), z2 = (z2, w2) ∈ HT. Let v = w1 f w2 . Then  d (z1, z2) , if v ∈ {w1, w2}  H (2.8) d z , z ) = HT 1 2 min{dH(z1, z) + dH(z, z2): z ∈ Lh(v)} ,   if v∈ / {w1, w2} .

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In the first case, z1 and z2 belong to a common copy Hξ of H in HT. In the ∗ second case, v ∈ V (T), and there are ξ1, ξ2 ∈ ∂ T such that ξ1 f ξ2 = v and both zi lie above Lv in Hξi , whence any geodesic from z1 to z2 must pass through a (unique!) point z ∈ Lv. See [3, Figure 5]. Next, we recall the isometry group of HT(q, p). First, the locally compact group of affine transformations qn b (2.9) Aff( , q) = g = : n ∈ , b ∈ , gz = qnz + b H 0 1 Z R acts on H(q) by isometries and preserves the set of “slicing” lines Lk (k ∈ Z). Left Haar measure dg and its modular function δH = δH,q are −n −n qn b dg = q dn db and δH(g) = q , if g = 0 1 , where dn is counting measure on Z and db is Lebesgue measure on R. Second, denote by Aut(Tp) the full isometry group of Tp and consider the affine group of Tp , − − (2.10) Aff(Tp) = {γ ∈ Aut(Tp):(γv) = v for all v ∈ V (Tp)} .

The modular function δT = δTp of Aff(Tp) is given by Φ(γ) δT(γ) = p where Φ(γ) = h(γw) − h(w) , if γ ∈ Aff(Tp) , w ∈ T . The above mapping Φ: Aff(T) → Z is independent of w ∈ T and a homo- morphism.

Theorem 2.5 ([3]).— The group

A = A(q, p) = {(g, γ) ∈ Aﬀ(H, q) × Aﬀ(Tp) : logq δH(g) + logp δT(γ) = 0} acts by isometries (g, γ)(z, w) = (gz, γw) on HT(q, p), with compact quotient isomorphic with the circle of length log q. It leaves the area element dz of HT as well as the transition kernel (2.4) invariant:

(2.11) pα,β(t, gw, gz) = pα,β(t, w, z) for all t > 0 , w, z ∈ HT and g ∈ A .

Next, we recall from [3] the stopping times τ(n) (n ∈ N0) of the successive visits of (Yt) in Z (2.12) τ(0) = 0, τ(n + 1) = inf t > τ(n): Yt ∈ Z \{Yτ(n)} . Via Proposition 2.4, we can also interpret them in terms of Brownian mo- o tion on HT. If X0 lies in Sv, then τ(1) is the instant when Xt ﬁrst meets a point on Lv ∪ Lv− . If Xτ(n) ∈ Lv for some v ∈ V (T) (which holds for

TOME 66 (2016), FASCICULE 4 1700 A. Bendikov, L. Saloff-Coste, M. Salvatori & W. Woess all n > 1, and possibly also for n = 0), then τ(n + 1) is the first instant − t > τ(n) when Xt meets one of the bifurcation lines Lv− or Lw with w = v. The increments τ(n) − τ(n − 1), n > 1, are independent and almost surely finite. They are identically distributed for n > 2, and when Y0 ∈ Z (equivalently, Z0 ∈ LT), then also τ(1) has the same distribution. The random variables

(2.13) τ = τ(2) − τ(1) and Y = Yτ(2) − Yτ(1) are independent, a 1 (2.14) Pr[Y = 1] = and Pr[Y = −1] = , a = βp qα−1 , a + 1 a + 1

λ0τ and τ has finite exponential moment E(e ) for some λ0 > 0. For any open domain Θ ⊂ HT, we let Θ (2.15) τ = inf{t > 0 : Xt ∈ HT \ Θ} be the first exit time of (Xt) from Θ. Θ If τ = τ < ∞ almost surely for the starting point X0 = w ∈ Θ, then Θ we write µw for the distribution of Xτ . In all cases that we shall consider, Θ µw is going to be a probability measure supported by ∂Θ. We shall use analogous notation on H, T and R. We note that by group invariance of our Laplacian, Θ gΘ (2.16) µw(B) = µgw(gB) for every g ∈ A and Borel set B ⊂ HT . We conclude this section with the “official” definition of harmonic functions.

Definition 2.6. — Let Θ ⊂ HT be open. A continuous function f : Θ → R is called harmonic on Θ if for every open, relatively compact set U with U ⊂ Θ, Z U f(z) = f dµz for all z ∈ U.

As mentioned in [3], from a classical analytic viewpoint, “harmonic” should rather mean “annihilated by the Laplacian”, but for general open domains in HT, the correct formulation in these terms is subtle in view of the relative location of the bifurcations. By [2, Theorem 5.9], if Θ is suitably “nice”, then any harmonic function on Θ is anihilated by ∆α,β . This is true in particular for the sets Ωr that we are going to use in the next section; see Figure 2 below. More details will be stated and used later on; in particular, see Proposition 3.1 regarding globally harmonic functions on HT.

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3. Harmonic functions on rectangular sets

In this section, we want to derive a Poisson representation formula for harmonic functions on open domains Ω ⊂ HT which have a rectangular shape. Prototypes are the following sets, where v ∈ V (T) and r > 0. o Ωv = {(z, w) ∈ HT : w ∈ N(v) } ⊂ HT , and (3.1) Ωv,r = {z ∈ Ωv : |Re z| < r} , where N(v) is the “neighbourhood star” at v in T. That is, N(v) is the union of all edges (≡ intervals !) of T which have v as one endpoint. It is a compact metric subtree of T, whose boundary ∂N(v) consists of all neighbours of v in V (T). We write ∂+N(v) = ∂N(v) \{v−} (the forward neighbours of v). ∞ Note that we can deﬁne the space C (Ω) for Ω = Ωv or Ωv,r in the same way as in Deﬁnition 2.1 by restricting to the (pieces of the) strips that make up Ω . We recall the following [2, Theorem 5.9]:

Proposition 3.1. — A function f is harmonic on Ω = HT, resp. Ω = ∞ Ωv , resp. Ω = Ωv,r if and only if f ∈ C (Ω), it satisﬁes the bifurcation con- 2 2 2 ditions (2.3) at each bifurcation line inside Ω, and y (∂x+∂y )f +α y ∂yf = 0 in each open strip in Ω.

For deriving a Poisson representation for harmonic functions on Ωv , it is sufficient to consider just Ωo (by group invariance). We shall henceforth always reserve the letter Ω for this specific set. We follow classical reasoning, but here we have to be careful in view of the singularities of our domain along the bifurcation line Lo ; the validity of each step has to be checked. We first work with the bounded subsets Ωr = Ωo,r . Here, r > 0 will always be a real number, so that Ωr should not be confounded with Ωv for v ∈ V (T). The boundary ∂Ωr in HT consists of the horizontal part and the vertical part, given by hor ∂ Ωr = {(z, v) ∈ HT : v ∈ ∂N(o) , |Re z| 6 r} and vert (3.2) ∂ Ωr = Nr(o) ∪ N−r(o) , where for any r ∈ R ,

Nr(o) = {(z, w) ∈ HT : w ∈ N(o) , Re z = r} , respectively. The horizontal part lies on the union of all bifurcation lines Lv , v ∈ ∂N(o). The vertical part consists of two isometric copies of the compact metric tree N(o) within HT, which delimit our set on the left and right hand sides, at which Re z = −r, resp. Re z = r. We call the elements of the ﬁnite set (z, v) ∈ HT : |Re z| = r , v ∈ V N(o) the corners of Ωr .

TOME 66 (2016), FASCICULE 4 1702 A. Bendikov, L. Saloﬀ-Coste, M. Salvatori & W. Woess

...... ◦...... ◦...... ◦...... ◦... . ◦...... • ...... o ...... ◦ ...... ◦...... ◦......

Figure 2. The compact set Ωr . Its corners are encircled.

Our Ωr is the union of the ﬁnitely many closed “rectangles” Rv,r , where

o Rv,r = {z ∈ Sv : |Re z| < r}.

The boundary of Rv,r consists of two horizontal sides Lv,r and Lv−,r and two vertical sides Jv,r and Jv,−r , where Lv,r = {(z, v) ∈ Lv : |z| 6 r} and − Jv,r = {(z, w) ∈ HT : w ∈ [v , v] , Re z = r}. Here, the vertex v ranges in + {o} ∪ ∂ N(o), and the closed rectangles Rv,r and Ro,r meet at their lower, resp. upper horizontal sides, when v− = o. We want to write down Green’s formulas for “nice” functions on Ωr . We start by specifying those functions.

2 Definition 3.2. — Let Dα,β(Ωr) be the space of all continuous functions f on Ωr with the following properties. (i) For v = o as well as for any forward neighbour v of o in V (T), the partial derivatives of f up to 2nd order exist and are bounded and continuous on Rv,r , and (ii) they extend continuously from Rv,r to ∂Rv,r , except possibly at the corners of Ωr. (But on bifurcation lines, the extensions coming from diﬀerent strips will in general not coincide.) (iii) The ﬁrst order derivatives satisfy the bifurcation condition (2.3) along the segment Lo,r . (iv) ∆α,βf extends continuously to Ωr across the bifurcation line.

ANNALES DE L’INSTITUT FOURIER HARMONIC FUNCTIONS ON TREEBOLIC SPACE 1703

If U ⊂ Ωr is a “nicely shaped” open subset of Ωr, we define the space 2 2 Dα,β(Ωr \ U) analogously. Furthermore, we define Dα,β(Ω) as the space of 2 all functions on Ω whose restriction to Ωr is in Dα,β(Ωr) for every r > 0. Next, we need to specify the boundary (arc) measure induced by the measure m = mα,β of (2.2). (We shall omit the paramters α, β in the 0 0 index.) More precisely, we need one such arc measure mv = mα,β;v on each closed strip Sv , where v ∈ V (T), given by 0 p 2 2 (3.3) dmv(z) = φα,β(z) dx + dy y for z = (x + i y ; v) ∈ Sv , where φα,β(z) is as in (2.2). Contrary to the definition (2.2) of m, the lower boundary line Lv− of Sv is not excluded in this definition, so that on that 0 0 line we have the two arc measures mv and mv− which differ by a factor of β. Their use depends on the side from which the line is approached. Now we can construct the boundary measure m∂Ωr and the boundary gradient 2 of f ∈ Dα,β(Ωr) as follows:

+ ∂Ωr 0 ∂Ωr (1) for each v ∈ ∂ N(o), we let m = mv and ∇ f = ∇fv (the extension to ∂Ωr within Sv) on the boundary parts Lv,r and on Jv,±r ; ∂Ωr 0 ∂Ωr (2) furthermore, m = mo and ∇ f = ∇fo (the extension to ∂Ωr within So) on the boundary parts Lo−,r and on Jo,±r . In (1) and (2), the corners of Ω – a set with measure 0 – are excluded. In the following, the occuring boundaries are considered with positive ori- entation.

2 Proposition 3.3. — For f, h ∈ Dα,β(Ωr), Z Z ∂Ωr ∂Ωr f ∆α,βh + (∇f, ∇h) dm = (n, ∇ h) f dm and Ωr ∂Ωr Z Z ∂Ωr ∂Ωr ∂Ωr f ∆α,βh − h ∆α,βf) dm = (n, ∇ h) f − (n, ∇ f) h dm , Ωr ∂Ωr

∂Ωr where n(z) = n (z) is the outward unit normal vector to ∂Ωr at z ∈ ∂Ωr , defined except at the corners of Ωr . Proof. — We can write down the classical first Green formula as above ∂R on each R = Rv,r, with the boundary gradient ∇ f : Z Z ∂R 0 f ∆α,βh + (∇f, ∇h) dm = (n, ∇ h) f dmv . Rv,r ∂Rv,r ∂R Here, as above for Ωr above, the boundary gradient ∇ f is the continuous extension of the gradient to ∂Rv,r , well defined except at the corners.

TOME 66 (2016), FASCICULE 4 1704 A. Bendikov, L. Saloﬀ-Coste, M. Salvatori & W. Woess

Indeed, we can first inscribe a slightly smaller rectangle whose closure is contained in Rv,r . We then have Green’s first formula on that set, as stated in any calculus textbook. Then we can increase that inscribed rectangle and pass to the limit. The exchange of limit and integrals is legitimate because of the boundedness assumptions that appeared in the definition of the space 2 Dα,β(Ω). Now we take the sum of those integrals over all Rv,r that make up Ωr . The segment Lo,r of the bifurcation line Lo lies in the interior of Ω. The bifurcation condition makes sure that the line integrals over Lo,r coming from the different adjacent rectangles sum up to 0. Green’s second formula follows from the first one. By [2, Theorems 5.9 and 5.19], the set Ωr admits a Dirichlet Green kernel Ωr 2 g (·, w) which for w ∈ Ωr belongs to Dα,β(Ωr \U) for any open neighbourhood U of its pole w. The fact that gΩr (·, w) is smooth up to the boundary segments of our Ωr (with exception of the corners, where, however, the partial derivatives are bounded) follows from the general theory of ellptic 2nd order PDEs; see Evans [18, Theorem 6, p. 326]. We use it to define the Poisson kernel Ωr ∂Ωr ∂Ωr Ωr (3.4) Π (w, z) = − n (z) , ∇z g (z, w) , w ∈ Ωr , z ∈ ∂Ωr . (The gradient is applied to the variable z in the index, or rather to its hyperbolic part z = x + i y).) Note that the Poisson kernel is positive. The argument used for the next lemma is classical, but it is important to check that each step works in our “non-classical” setting with the bifurcation lines and Kirchhoff condition.

2 Lemma 3.4 (Poisson representation).— Let h ∈ Dα,β(Ωr) be harmonic, that is, ∆α,βh = 0 in Ωr . Then for every w ∈ Ωr , Z h(w) = ΠΩr (w, z)h(z) dm∂Ωr (z) . ∂Ωr ∞ Proof. — Let ϕ be any non-negative function in Cc (Ωr) and let f be its Green potential on Ωr , that is, Z (3.5) f(w) = gΩr (w, z) ϕ(z) dm(z) . Ωr Then f is a weak solution of the equation

∆α,βf = −ϕ in Ωr .

Smoothness of the heat kernel hα,β(t, w, z) associated with ∆α,β (see [2, Theorem 5.23 and Appendix 1]) implies as in the classical theory of PDE

ANNALES DE L’INSTITUT FOURIER HARMONIC FUNCTIONS ON TREEBOLIC SPACE 1705 that f is a strong solution of the abvove Laplace equation, that is, f ∈ 2 Dα,β(Ωr) (indeed, it has all higher order derivatives). We have f = 0 on ∂Ωr. Inserting f and h into Green’s second identity of Proposition 3.3, we get with an application of Fubini’s theorem Z Z ∂Ωr ∂Ωr h(w) ϕ(w) dm = − n(z), ∇z f(z) h(z) dm (z) Ωr ∂Ωr Z Z ∂Ωr Ωr ∂Ωr = n(z), ∇z g (w, z) ϕ(z) h(z) dm(w) dm (z) ∂Ωr Ωr Z Z = ΠΩr (w, z)h(z) dm∂Ωr (z) ϕ(w) dm(w) . Ωr ∂Ωr Since this holds for any continuous function ϕ as speciﬁed above, the statement follows.

Let f be a continuous function on ∂Ωr . Then Z Ωr h(w) = f dµw = Ew f(Xτ Ωr ) ∂Ωr is the unique solution of the Dirichlet problem with boundary data f. By Lemma 3.4, Z h(w) = ΠΩr (w, z)f(z) dm∂Ωr (z) . ∂Ωr

Ωr Corollary 3.5. — For any w ∈ Ωr , the Poisson kernel Π (w, ·) is Ωr the density of the hitting (≡ exit) distribution µw with respect to the boundary measure m∂Ωr .

We want to have an analogous result for the non-compact set Ω = Ωo . In spite of being “obvious”, this requires further substantial work.

Lemma 3.6. — For any w ∈ Ω and any r > |Re w|,

Ωr vert Ω r−|Re w|−1 µw ∂ Ωr 6 Prw max{|Re Xt| : t 6 τ } > r 6 2ρ , where ρ < 1.

Proof. — The ﬁrst inequality is clear: if up to the exit time from the entire set Ω we have |Re Xt| 6 r then Xt cannot exit Ωr through its vertical boundary part. For the second inequality, the arguments are those used in the proof of [3, Proposition 4.18]. In particular, see [3, Figure 7]. The number ρ is as in that proposition. The result there is formulated in terms of the projected process on H(q), and we add some brief reminders in terms of HT itself.

TOME 66 (2016), FASCICULE 4 1706 A. Bendikov, L. Saloﬀ-Coste, M. Salvatori & W. Woess

We “slice” Ω vertically by the sets N±k(o), k ∈ N, and let σ(k) be the Ω exit time of our Brownian motion from Ωk, while τ coincides with the σ of [3, (4.3)]: with respect to the projection Yt = πR(Xt) on the real line, Ω τ = inf{t > 0 : Yt ∈/ [−1, 1]} = inf{t > 0 : Yt = ±1} , (3.6) where Y0 ∈ [−1 , 1], which is a.s. ﬁnite and has an exponential moment. We also note that Ω σ(k) 6 τ . hor The function z 7→ Prz[Xσ(1) ∈ ∂ Ω1] is strictly positive and weakly harmonic (≡ in the sense of distributions) on Ω1 , whence strongly harmonic by [2, Theorem 5.9], and consequently continuous. Along N0(o), it must attain its minimum in some z∗ ∈ N0(o) \ ∂Ω1 . Then hor ρ = 1 − Prz∗ [Xσ(1) ∈ ∂ Ω1].

Now let us assume first that w ∈ Ω is such that Re w = 0. If Xt starts at Ω w and max{Re Xt : t 6 τ } > r, then it must pass through all the vertical “barriers” N1(o) , ··· ,Nn(o), where n = brc. But by group-invariance (just using horizontal translations here), for any starting point in Nk−1(o), the probability that Xt reaches Nk(o)\∂Ω before ∂Ω is bounded above by ρ. By a simple inductive argument as in in the proof of [3, Proposition 4.18], the n r−1 probability to pass through all those Nk(o) before exiting Ω is 6 ρ 6 ρ . Ω r−1 In the same way, Prw min{Re Xt : t 6 τ } < −r 6 ρ . Thus, the stated upper bound of the second inequality holds when Re w = 0. If w ∈ Ω is arbitrary, then we can use group invariance and map w to a point (0, w0) by a horizontal translation of HT. Then we must replace Ωr by Ωr−|Re w| in the preceding arguments. Since τ Ω = σ is a.s. finite, also the set Ω has a Dirichlet Green ker- Ω 2 nel g (·, w) which for w ∈ Ω belongs to Dα,β(Ω \ U) for every “nicely 2 shaped” open neighbourhood U of its pole w. Here, the space Dα,β(Ω) is as in Definition (3.2), except that we do not have to refer to the corners, and boundedness of partial derivatives is local, i.e., it refers to arbitrary relatively compact subsets of any of the strips that make up Ω. The following is another relative of [3, Proposition 4.18] and the above Lemma 3.6.

Lemma 3.7. — For any w ∈ Ω, gΩ(z, w) → 0 as d(z, w) → ∞ , z ∈ Ω .

Proof. — We use classical balayage, see Dynkin [14, (13.82) in §13.25] and Blumenthal and Getoor [4, Theorem (1.16) on p. 261]. We have for all

ANNALES DE L’INSTITUT FOURIER HARMONIC FUNCTIONS ON TREEBOLIC SPACE 1707 z, w ∈ Ω and r > |Re w| Z Z Ω Ωr Ω Ωr Ω Ωr (3.7) g (w, z) − g (w, z) = g (·, z) dµw = g (·, z) dµw , vert ∂Ωr ∂ Ωr Ω hor where the last equality holds because g (·, z) = 0 on ∂ Ωr . Of course, we Ωr have g (w, z) = 0 unless also z ∈ Ωr . For ﬁxed w, when d(z, w) is suﬃciently large, there is r = r(z) such that w ∈ Ωr and z ∈ Ω \ Ωr , and r(z) → ∞ when d(z, w) → ∞ . Then

Ω Ω Ωr vert Ω vert g (z, w) = g (w, z) 6 µw ∂ Ωr × sup{g (u, z): u ∈ ∂ Ωr}. By Lemma 3.6, this tends to 0, as r = r(z) → ∞ . We can now deﬁne the Poisson kernel ΠΩ(w, z) as in (3.4). It is posi- Ω Ω tive and supported by ∂Ω . We have Π (w, z) = ∂yg (z, w) on Lo− and Ω Ω − Π (w, z) = −∂yg (z, w) on each Lv with v = o. (The partial deriviative ∂y refers to the y-coordinate of z = x + i y.) Proposition 3.8. Z ΠΩ(w, z) dm∂Ω = 1 . ∂Ω Proof. — For r > 0, we use Green’s second identity of Proposition 3.3 ∞ on Ωr . We again let ϕ be any non-negative function in Cc (Ωr). We use the Green potential of ϕ as in (3.5), but this time with respect to the whole of Ω instead of Ωr . Furthermore, we let h ≡ 1. We have f = 0 on ∂Ω, in hor particular on ∂ Ωr , and ∆α,βf = −ϕ on Ω. Thus, we get Z Z Ωr ∂Ωr ∂Ωr ϕ(w) dm = − n (z) , ∇z f(z) dm (z). Ωr ∂Ωr

We ﬁx w ∈ Ωr . Continuing as in the proof of Lemma 3.4, we see that Z Ωr ∂Ωr Ω ∂Ωr 1 = − n (z) , ∇z g (z, w) dm (z) ∂Ωr Z = ΠΩ(w, z) dm∂Ωr (z) hor ∂ Ωr Z Z 2 Ω ∂Ωr 2 Ω ∂Ωr + y ∂xg (z, w) dm (z) + −y ∂xg (z, w) dm (z) , N−r Nr where N−r = N−r(o) and Nr = Nr(o) are the left and right hand vertical boundary parts of Ωr . Since we have ﬁxed w, we can write gΩ(z, w) = g(z) = g(x+i y, w), when z = (x + i y, w) with w ∈ Tv . We observe that on Nr for arbitrary r ∈ R,

Ωr dlog (y)e α dm (z) = β q y dy , where z = (r + i y, w) ∈ Nr . | {z } =: φ(y)

TOME 66 (2016), FASCICULE 4 1708 A. Bendikov, L. Saloﬀ-Coste, M. Salvatori & W. Woess

The derivative ∂x refers to the x-coordinate of z. We can consider the last measure, as well as the next integral below, as a measure and integral on the neighbourhood star N(o) in T, because Nr is an isometric copy of that neighbourhood star. Thus, for r ∈ R, we can write Z Z 2 Ω ∂Ωr 2 y ∂xg (z, w) dm (z) = y ∂xg(r + i y, w) φ(y) dy. Nr N(o) This notation is slightly inprecise; we keep in mind that in the integral over N(o), one has to take the sum over such integrals, where each one ranges over one of the edges of N(o) . We have obtained that for any x ∈ R \{0}, Z 2 1 − Π(|x|) = y ∂x g(−|x| + i y, w) − g(|x| + i y, w) φ(y) dy N(o) Z where Π(|x|) = ΠΩ(w, z) dm∂Ω|x| (z) . hor ∂ Ω|x| Now we choose a ﬁxed ε > 0 and let r > ε. We integrate both sides over x which varies in the interval I = (r −ε , r +ε). Then we get, with an obvious exchange of the order of integration Z 2ε − Π(|x|) dx I Z Z 2 = y ∂x g(−|x| + i y, w) − g(|x| + i y, w) dx φ(y) dy N(o) I Z = g(−r − ε + i y, w) − g(r + ε + i y, w) N(o) − g(−r + ε + i y, w) + g(r − ε + i y, w) φ(y) dy

When r → ∞, the last integral tends to to 0 by Lemma 3.7, while the R Ω ∂Ω one on the left hand side tends to 2ε ∂Ω Π (w, ·) dm . Thus, Z 2ε = 2ε ΠΩ(w, z) dmΩ(z) , ∂Ω which yields the proposed statement. We are ﬁnally in the position to extend Lemma 3.4 to the unbounded set Ω = Ωo in HT. This will also prove that the exit measure µ from Ω has a continuous density with respect to Lebesgue measure on ∂Ω, or equivalently, with respect to Haar measure on A.

ANNALES DE L’INSTITUT FOURIER HARMONIC FUNCTIONS ON TREEBOLIC SPACE 1709

2 Theorem 3.9 (Poisson representation).— Let h ∈ Dα,β(Ω) be harmonic, that is, ∆α,βh = 0 in Ω. Suppose that h satisﬁes the growth condition

C2 dHT(z,o) (3.8) |h(z)| 6 C1 e on Ω (C1 ,C2 > 0). Then for every w ∈ Ω, Z h(w) = ΠΩ(w, z)h(z) dm∂Ω(z) . ∂Ω In particular, the Poisson kernel ΠΩ(w, ·) is the density of the exit distri- Ω ∂Ω bution µw with respect to the boundary measure m , where w ∈ Ω. Proof. — We ﬁrst assume that h is bounded. Let r be large enough such Ωr Ωr ∂Ωr that w ∈ Ωr . Then by Lemma 3.4 and because dµw = Π (w, ·) dm , Z h(w) = ΠΩr (w, z)h(z) dm∂Ωr (z) ∂Ωr Z Z Ωr ∂Ω Ωr = Π (w, z)h(z) dm (z) + h(z)dµw (z) . hor vert ∂ Ωr ∂ Ωr vert When r → ∞, the last integral along ∂ Ωr tends to 0 by Lemma 3.6 and boundedness of h. Z Z ΠΩr (w, z)h(z) dm∂Ω(z) → ΠΩ(w, z)h(z) dm∂Ω(z) . hor ∂ Ωr ∂Ω

Taking normal derivatives in (3.7), we have for all w ∈ Ωr and z ∈ ∂Ω Z Ω Ωr Ω Ωr Π (w, z) − Π (w, z) = Π (·, z) dµw > 0 vert ∂ Ωr

Ωr hor with Π (w, z) = 0 when z ∈ ∂Ω\∂ Ωr . Therefore, using Proposition 3.8, Z Z Ω ∂Ω Ω ∂Ω Π (w, z)h(z) dm (z) − Π r (w, z)h(z) dm (z) ∂Ω ∂Ω Z Z Ω ∂Ω Ωr Ωr vert = Π (·, z)h(z) dm (z) dµw 6 khk∞ µw ∂ Ωr , ∂vertΩ ∂Ω which tends to 0 as r → ∞ by Lemma 3.6. At this point, we have obtained the Poisson representation for all bounded harmonic functions. In the same way as we got Corollary 3.5, we get that

Ω ∂Ω Ω (3.9) Π (w, z) dm (z) = dµw(z) , as proposed.

TOME 66 (2016), FASCICULE 4 1710 A. Bendikov, L. Saloﬀ-Coste, M. Salvatori & W. Woess

Before we proceed, we make a simple observation on the metric. For any z = (z, w) ∈ HT,

dHT(z, o) 6 2 log(|Re z| + 1) + dT(w, o) log q. Indeed, this is the triangle inequality: let z0 = (Re z + i , o) be the point 0 0 0 on Lo with Re z = Re z. Then dHT(z, z ) = dT(w, o) log q, and dHT(z , o) = dH(Re z +i , i ) 6 2 log(|Re z|+1), a simple computation with the hyperbolic metric. Thus,

(3.10) dHT(z, o) 6 2 log(|Re z| + 1) + D, for all z ∈ Ω , where D = log q . Now let h be an arbitrary harmonic function on Ω that satisﬁes (3.8). Suppose ﬁrst thatRe w = 0. Let σ(0) = 0, and as in the proof of Lemma 3.6, let σ(k) be the exit time of Brownian motion from Ωk, where k ∈ N. Then harmonicity of h on Ω yields that h(Xσ(k)) is a discrete-time k>0 martingale. Consider the random variable

(3.11) Rmax = max{|Re Xt| : t 6 τ}. By Lemma 3.6, we have that

s Ez(Rmax) < ∞ for every s > 0. Combining the growth condition (3.8) with (3.10), we get

2C C2D 2 |h(Xt)| 6 C1 e Rmax + 1 for 0 6 t 6 τ , an integrable upper bound. In particular, our martingale converges almost surely. hor We now note that σ(k) = τ if (Xt) exits Ωk at ∂ Ωk and that σ(k) < τ vert if (Xt) exits Ωk at ∂ Ωk . In particular, σ(k) = τ when k > Rmax . We can decompose

h(Xσ(k)) = h(Xτ ) 1[σ(k)=τ] + h(Xσ(k)) 1[σ(k)<τ] , and see that h(Xσ(k)) → h(Xτ ) almost surely. Using dominated convergence and the martingale property, we conclude that f(z) = Ez h(Xτ ) , which proves the required statement for the case when Re z = 0. For arbir- tary z ∈ Ω, it follows immediately from the horizontal translation invariance of our Laplacian.

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Ω Corollary 3.10. — For any w ∈ Ω, the exit measure µw has continuous, strictly positive density with respect to Lebesgue measure on the lines that make up ∂Ω. It satisﬁes the moment condition Z Ω exp λ |Re z| dµw(z) < ∞ ∂Ω for any λ < log(1/ρ) with ρ as in Lemma 3.6. Thus, we have the double exponential moment condition, Z Ω exp exp dHT(o, z)/3 dµw(z) < ∞ . ∂Ω Ω Proof. — The fact that supp µw = ∂Ω follows from (3.9), or equivalently from the observation made in the proof of the next corollary that all boundary points are regular: indeed, every regular point must belong to the support of the exit distribution. The moment condition follows from Lemma 3.6: if (Xt) starts at w ∈ Ωr and exits Ω at some z ∈ ∂Ω \ ∂Ωr then it must exit Ωr through its vertical boundary, that is

Ω Ωr vert µw(∂Ω \ ∂Ωr) 6 µw ∂ Ωr . Theorem 3.11 (Solution of the Dirichlet problem).— Let f ∈ C(∂Ω) be such that the growth condition (3.8) holds for f on ∂Ω. Then Z Ω ∂Ω  Π (w, z)f(z) dm (z) , w ∈ Ω , h(w) = ∂Ω f(w) , w ∈ ∂Ω deﬁnes an extension of f that is harmonic on Ω and continuous on Ω. Furthermore, there is at most one such extension which satisﬁes (3.8) on the whole of Ω.

Proof. — A boundary point z of any open domain Ω ⊂ HT is regular for the Dirichlet problem for continuous functions on ∂Ω if and only if the Ω exit distribution satisﬁes Prz[τ = 0] = 1 (a general fact from Potential Theory). Every boundary point of Ω is regular. This follows from the fact that τ Ω (which was denoted σ above) is the same as the exit time of the projected process (Yt) on R from the interval [−1 , 1], see (3.6). But the Dirichlet problem for the latter interval (with boundary values at ±1) is obviously solvable, as veriﬁed by direct, elementary computations; see [3, Lemma 4.4 & proof].

TOME 66 (2016), FASCICULE 4 1712 A. Bendikov, L. Saloﬀ-Coste, M. Salvatori & W. Woess

Now, the function h(w) defined in the corollary is harmonic; its finiteness is guaranteed by (3.8) in combination with Corollary 3.10. By the same reason, it is continuous up to ∂Ω, since every boundary point is regular, as we have just observed. Thus, we have a solution of the Dirichlet problem. Since Ω is not bounded, uniqueness is not immediate. (In the compact case, uniqueness would follow from the maximum principle: a harmonic function assumes its maximum on the boundary.) However, if some extension satisfies (3.8) then we can apply Theorem 3.9 to see that it must coincide with the function defined in the present theorem. Remark 3.12. — What we have stated and proved for Ω also works in precisely the same way, with some more notational efforts, for more general sets which are unions of finitely many strips. We state several facts that will be used in the sequel. (a) As already mentioned, group invariance implies that Theorem 3.9, Theorem 3.11 and Corollary 3.10 are valid for any set Ωv (v ∈ V (T)) in the place of Ω = Ωo .

(b) We can take a compact metric subtree T of Tp which is full in the sense that for each of its vertices, either all, or else just one of its neigbours is in T . Then we can take the set

ΩT = {z = (z, w) ∈ HT : w ∈ T },

and ΩT is its interior. By precisely the same methods as those used above, with only minor and straightforward notational adaptations, one shows that also in this case, the exit distribution from ΩT (with respect to any starting point in that set) has a continuous density with respect to the Lebesgue measure on the union of the bifurcation lines that make up ∂ΩT . We also have the analogous Poisson representation and solution of the Dirichlet problem on ΩT , as well as the exponential moment for the exit distribution, which is supported by the entire boundary of ΩT . (c) For any set Θ ⊂ HT and ε > 0, we let (3.12) Θε = {z ∈ Θ: d(z, ∂Θ) > ε}. ε For all results stated in (b), we can also replace ΩT by ΩT , where ε 0 < ε < log q : thus, ∂ΩT consists of horizontal lines in the interior of the boundary strips of ΩT at (hyperbolic) distance ε from the boundary lines of ΩT . (d) The same holds for “sliced” hyperbolic plane, interpreted as HT(q,1). In this case, Ωv should be replaced by the double strip Ωek = πH(Ωv)

ANNALES DE L’INSTITUT FOURIER HARMONIC FUNCTIONS ON TREEBOLIC SPACE 1713

with k = h(v), and more generally this holds for any union of finitely many strips which is connected in H, and we may extend those strips (or truncate) above and below up to some intermediate boundary lines. (e) In particular, all those results also hold for a single strip. Note that this is in reality a classical situation because before reaching the boundary of a strip, we are just observing ordinary hyperbolic Brownian motion with drift. The reason why we did not state any more general results, e.g. for relatively compact domains that do not have rectangular shape, lies in the complication of how those sets may cross (several times from different sides) the bifurcation lines. For sets with sufficiently regular shape and piecewise smooth boundaries, such results can be stated and elaborated with consid- erable further notational efforts, while the techniques remain basically the same.

4. Positive harmonic functions and the induced random walk

We now ﬁnally turn our attention to positive harmonic functions on the whole of HT. Recall the stopping times τ(n) from (2.12). We assume that Brownian motion starts in some point of LT, so that all increments τ(n) − τ(n − 1), n > 1, are i.i.d. and have the same distribution as τ(1). Before coming to the core of this section, we start with a simple result concerning the induced random walk (Wτ(n))n>0 on T. An easy calculation [3, Cor. 4.9] shows that when it starts at a vertex of T, then this is a nearest neighbour random walk on the vertex set of T with transition probabilities 1 a (4.1) p (v, v−) = and p (v−, v) = , v ∈ V ( ) T 1 + a T (1 + a)p T with a as in (2.14). The one-step transition probabilities beween all other pairs of vertices are 0.

Lemma 4.1. T (i) If a non-negative function f on T is ∆α,β-harmonic then its restriction to V (T) is harmonic for the transition probabilites of (Wτ(n)). That is, for any v ∈ V (T) X (4.2) f(v) = pT(v, u) f(u) . u∈V (T): u∼v

TOME 66 (2016), FASCICULE 4 1714 A. Bendikov, L. Saloﬀ-Coste, M. Salvatori & W. Woess

(ii) Conversely, for any non-negative function f on V (T) which satis- T ﬁes (4.2), its unique extension to a ∆α,β-harmonic function on T is given on each edge [v−, v] of T and w ∈ [v−, v] by f(w) = λ(w) · f(v−) + 1 − λ(w) · f(v) with  q(α−1)(h(v)−h(w)) − 1 (4.3)  , if α 6= 1 , λ(w) = q(α−1) − 1 h(v) − h(w) , if α = 1 ,

where (recall) [v− , v] corresponds to the real interval [h(v)−1 , h(v)].

Proof. — (i) is clear from Deﬁnition 2.6 of harmonic functions. (ii) Given f on V (T) and any vertex, we see from (2.6) that the extension to T within any edge [v−, v] must satisfy the boundary value problem 1 00 α−1 0 − (log q)2 f + log q f = 0, with the given values f(v ) and f(v) at the end- points of the interval corresponding to that edge. Straightforward computation leads to the solution (4.3). It must satisfy the bifurcation condition of (2.6), which can also be veriﬁed directly.

After this warmup, our main focus is on the induced process (Xτ(n))n>0 on LT. We want to relate its positive harmonic functions with the positive harmonic functions for our Laplacian ∆ = ∆α,β on HT. Here, a function f : LT → R is harmonic for that process if f(z) = Ez f(Xτ(1) for any z ∈ LT. In view of the previous section, this means that Z Ωv (4.4) f(z) = f dµz for every v ∈ V (T) and z ∈ Lv . ∂Ωv The analogue of Lemma 4.1 is by no means as straightforward; the reason is that on HT, the strips are non-compact so that one can make long sideways detours in between two successive bifurcation lines. Let us deﬁne the probability measure

Ω [ (4.5) µ = µo on ∂Ω = Lv . v:v∼o Group invariance yields that for any z = (z, v) ∈ LT and g ∈ A with go = z, Ωv we have the convolution identity µz = δg ∗ µ. Since A acts transitively on LT, the transition probabilities of the induced process are completely determined by µ, so that we can consider it as the random walk on LT with law µ. In view of this, we call the harmonic functions of (4.4) the µ-harmonic functions on LT. They are necessarily continuous, since µ has a continuous density.

ANNALES DE L’INSTITUT FOURIER HARMONIC FUNCTIONS ON TREEBOLIC SPACE 1715

We note here that the measure µ is invariant under the stabiliser Ao of our reference point o in A. For sliced hyperbolic plane, i.e., when p = 1, the group is Aﬀ(H, q) and that stabiliser is trivial. When p > 2, the stabiliser is the compact subgroup of A consisting of all g = (idH, γ), where γ ∈ Aﬀ(T) with γo = o. We also note the following fact that was anticipated in [3]: by Theo- rem 3.9, the measure µ has a continuous density with respect to Lebesgue measure on the union of the boundary lines Lv , v ∼ o. Also, by Lemma 3.6 it has exponential moment with respect to the Euclidean distance on each of those lines. For positive harmonic functions, the following uniform Harnack inequality is a consequence of [2, Theorem 4.2].

Proposition 4.2. — For every non-negative ∆α,β-harmonic function h on (the whole of) HT and each d > 0, 0 0 h(z ) 6 Cd h(z) , whenever dHT(z, z ) 6 d , where Cd > 1 is such that Cd → 1 when d → 0. In particular, every non-negative harmonic function must be strictly positive in every point. The next lemma will be important for understanding the relation between ∆-harmonic functions on HT and µ-harmonic functions on LT. It makes use of the straightforward extensions of Theorem 3.9 ε and Theorem 3.11 to sets of the form ΩT and ΩT for a full subtree T of T, as clariﬁed in Remark 3.12. Lemma 4.3. — Let T be a full (compact) subtree of T. For every d > 0 and 0 < ε < log q there is a constant Md = Md,T,ε > 0, with Md → 1 as d → 0, such that for any positive harmonic function h on ΩT , 0 0 ε 0 h(z ) 6 Md h(z) , whenever z, z ∈ ΩT with |Re z − Re z| 6 d .

In particular, in this situation, the exit distributions from the set ΩT satisfy ΩT ΩT µz0 6 Md µz .

Proof. — In analogy with the set Ωv,r of (3.1), let us first take ΩT,r = ε {z ∈ ΩT : |Re z| < r} and its subset ΩT,r . The intrinsic metric of ΩT is equivalent with the naturally defined Euclidean metric of that set (i.e., the metric has to be extended across bifurcation lines using straight geodesics). Then it follows once more from [2, Theorem 4.2] that every positive har- 0 monic function on ΩT,r satisfies a Harnack inequality as proposed for z, z ε ε in ΩT,r . Note that the whole of ΩT , resp. ΩT can be covered by translates ε of ΩT,r , resp. ΩT,r . By group invariance of our Laplacian, that inequality holds on each translate with the same constants Md .

TOME 66 (2016), FASCICULE 4 1716 A. Bendikov, L. Saloﬀ-Coste, M. Salvatori & W. Woess

For the next Harnack inequality, we need some of the features of the geometry of HT, T and H, as outlined above after Proposition 2.4.

Proposition 4.4. — There are constants C1 ,C2 > 0 such that for every non-negative µ-harmonic function f on LT and all z, z0 ∈ LT,

0 0 C2 dHT(z,z ) f(z ) 6 C1 e f(z) . Proof. — Step 1. Recall that Ω is the star of strips in LT with middle line Lo . Since µ has a continuous density, the assumption of µ-harmonicity Ω yields that f is continuous, and it is µz -integrable on ∂Ω for every z ∈ Ω. Therefore Z Ω h(z) = f dµz ∂Ω deﬁnes a positive ∆-harmonic extension of f to the closure of Ω. Since f is µ-harmonic, h also coincides with f on the middle line Lo of Ω. The function h satisﬁes the Harnack inequality of Lemma 4.3. This yields the following for h and consequently for f.

0 0 2 0 If z, z ∈ L0 and |Re z − Re z | 6 2q then f(z ) 6 M f(z) , where M = M2q2 . For any v ∈ V (T), we can map Ω to the star of strips k Ωv with middle line Lv by a mapping g = (g, γ) ∈ A such that gz = q z with k = h(v), and γ ∈ Aﬀ(T). It preserves both ∆- and µ-harmonicity. That mapping dilates horizontal Euclidean distances by the factor of qk. Therefore we also have the following statement for any v ∈ V (T) with the same constant M.

0 0 k+2 0 (4.6) If z, z ∈ Lv and |Re z − Re z | 6 2q then f(z ) 6 M f(z) .

Step 2. We now use this for V (T) 3 v ∼ o and the associated point v = h(v) (i q , v) ∈ Lv ⊂ ∂Ω, an endpoint of the “star” N0(o) as deﬁned in (3.2). 2+h(v) 2+h(v) Let Iv = {z ∈ Lv : |Re z| 6 2q } = Lv,r with r = 2q . (Note that h(v) = ±1.) Combining the assumption of µ-harmonicity with (4.6), Z Z f(o) = f dµ > f dµ > µ(Iv) f(v)/M. ∂Ω Iv

By Corollary 3.10, µ(Iv) > 0. Re-using (4.6) on each Lv , [ (4.7) if z ∈ Lo ∪ Lv and |Re z| 6 2q then f(z) 6 M f(i ) , v∼o where M = M/ min{µe(Iv): v ∼ o}.

ANNALES DE L’INSTITUT FOURIER HARMONIC FUNCTIONS ON TREEBOLIC SPACE 1717

Step 3. Now we let u ∈ V (T) and z ∈ Lu. Set k = h(u), Then we can map i to z by a mapping g = (g, γ) ∈ A, where g = pk Re z ∈ Aﬀ( , q) 0 1 H and γ ∈ Aﬀ(T) is such that γo = v. Thus, (4.7) transforms into

0 [ 0 k+1 If z, z ∈ Lu ∪ Lv and |Re z − Re z | 6 2q (4.8) v∼u 0 then f(z ) 6 Mf f(z) , where Mf = 2M.

Step 4. Now let z = (z, v), z0 = (z0, v0) ∈ LT, and consider the situation 0 0 where vfv = v, that is, v lies “above” v or is = v. Then we can choose some + 0 ξ ∈ ∂ T such that v ∈ $ ξ and z, z ∈ Hξ , as outlined in the description of the metric (2.8) of HT. A geodesic arc z z0 (depending on ξ) is then given 0 (2) by the (isometric) image in Hξ of the geodesic arc z z in H. The arc z z0 has a highest point ˜z = (˜z, w˜), where z˜ = z ∧ z0. Our arc may cross more than one strip. It meets LT ∩ Hξ in successive points zj = (zj, vj), 0 j = 0, . . . , n, where z0 = z and zn = z . All vj are vertices on the ray v ξ in T. Except possibly for one “top” sub-arc that contains ˜z, any zj−1 zj − crosses one strip: in the initial, ascending part (if it is present), vj = vj−1 and that strip is Svj , while in the terminal, descending part (if present), − vj = vj−1 and that strip is Svj−1 . In both cases (i.e. except possibly for the top arc), dHT(zj−1 , zj) > log q (the distance between any two adjacent bifurcation lines), and we get

0 dHT(z, z ) > (n − 1) log q .

For any j ∈ {1, . . . , n}, let k(j) = h(vj−1 f vj) be the level (horocycle number) of the lower boundary line of the strip in which the j-th sub-arc is contained. Therefore, the H-coordinates of zj−1 and zj satisfyIm zj−1 ∧zj 6 qk(j)+1. Thus by (2.7)

k(j)+1 |Re zj − Re zj+1| = |Re zj − Re zj+1| 6 2q , and (4.8) yields that f(zj+1) 6 Mf f(zj). Putting the sub-arcs together,

0 n 0 log Mf f(z ) Mf f(z) exp C2 dHT(z, z ) + 1 f(z) , where C2 = 6 6 log q

(2) This explains that geodesics in HT are in general not unique, because there are diﬀerent feasible choices for ξ that may lead to diﬀerent geodesics, according to the 0 mutual position of z and z in H.

TOME 66 (2016), FASCICULE 4 1718 A. Bendikov, L. Saloﬀ-Coste, M. Salvatori & W. Woess

0 0 0 0 Step 5. At last, let z = (z, v), z = (z , v ) ∈ LT be such that v f v ∈/ {v, v0}. Then (2.8) shows that any geodesic arc z z0 in HT decomposes into two sub-arcs z z00 and z00 z0, each of which is as in Step 4. Therefore 0 0 00 00 0 f(z ) 6 exp C2 dHT(z , z ) + 1 f(z ) 6 exp C2 dHT(z , z) + 2 f(z) This concludes the proof. Now consider a set ΩT , where T is a full subtree of T, as deﬁned in Remark 3.12(b). We note the following facts.

Remark 4.5. — If the starting point z of (Xt) lies on a line Lv in the interior of ΩT (that is, v is an interior vertex of T ) then the exit distribution ΩT µz of (Xt) from ΩT is the same as the exit distribution of the random walk (Xτ(n)) from ΩT . In view of Remark (3.12), it has the same properties as those of lem- mas 3.6 (with diﬀerent ρ) and 3.7, as well as Proposition 3.8 and corollar- ies 3.10 and 3.11. Then we have the following (easier) analogue of Theorem 3.9.

Proposition 4.6. — Let f be a µ-harmonic function on ΩT , that is, it satisﬁes (4.4) on every line Lv ⊂ ΩT . Suppose furthermore that f satisﬁes the growth condition

C2 dHT(z,o) (4.9) |f(z)| 6 C1 e on ∂ΩT , where C1 ,C2 > 0. Then for every z ∈ ΩT ∩ LT, Z ΩT f(z) = f dµz . ∂ΩT Proof. — We ﬁrst observe that (3.10) from the proof of Theorem 3.9 also holds on ΩT with constant D = max{dT(w, o): w ∈ T }· log q . ΩT This time, let τ = τ be the exit time from ΩT . Let the starting point T be z ∈ ΩT ∩ LT . Consider the process Xn = Xmin{τ(n),τ} on ΩT ∩ LT . This is the induced random walk on LT stopped upon reaching ∂ΩT . Then T f(Xn ) is a martingale by µ-harmonicity of f. Note that τ = τ(n) for n>0 some random n. Thus, for non-random n tending to ∞, f(XT ) = f(X ) 1 + f(X ) 1 → f(X ) almost surely. n τ [τ6τ(n)] τ(n) [τ>τ(n)] τ As in (3.11), we consider the random variable Rmax = max{|Re Xt| : t 6 τ} (only that this time, τ is more general). By Lemma 3.6, resp. its variant for ΩT , we have again that it has ﬁnite moments of all orders. Combining the growth condition (4.9) with (3.10), we get 2C T C2D 2 |f(Xn )| 6 C1 e Rmax + 1 ,

ANNALES DE L’INSTITUT FOURIER HARMONIC FUNCTIONS ON TREEBOLIC SPACE 1719 an integrable upper bound. Thus, by dominated convergence and the Mar- tingale property, f(z) = Ez f(Xτ ) , which is just another form of the proposed formula. Now we can formulate and prove the main result of this section, which is the analogue of Lemma 4.1 for HT.

Theorem 4.7.

(i) If a non-negative function h on HT is ∆α,β-harmonic then its restriction to LT is µ-harmonic. (ii) Conversely, for any non-negative µ-harmonic function f on LT, its unique extension to a ∆α,β-harmonic function on HT is given in o each open strip Sv of HT by

Z o Sv (4.10) h(z) = Ez f(Xτ(1)) = f dµz , ∂Sv

o Sv o where (recall) µz is the exit distribution from Sv of (Xt) starting o at z ∈ Sv .

Proof. — (i) If h is non-negative and is ∆α,β-harmonic on HT then it satisfies the Harnack inequality 4.2. Therefore, it also satisfies the growth condition (3.8), and Theorem 3.9 applies. That theorem provides (i) for points z ∈ Lo. By transitivity of the action of the group A on LT as well as group invariance of the Laplacian, this holds along any line Lv . (ii) We consider the following exhaustion of T by finite, full subtrees T (k). Let ok be the k-th predecessor of the root o of T, that is, the element on o $ at distance k from o. Let W (k) be the set of all vertices on the horocycle Hk at distance 2k − 1 from ok−1. (That is, they have ok−1 as their (2k − 1)-st predecessor.) Then T (k) is the union of all geodesic segments ok w, where w ∈ W (k). We consider ΩT (k) , with [ ∂ΩT (k) = Lok ∪ Lw . w∈W (k) Now let f be a positive µ-harmonic function on LT. Then it is continuous (since µ has a continuous density by Theorem 3.9) and satisfies (4.9) by Proposition 4.4. In particular, we can apply Theorem 3.11, resp., its extended version according to Remark 3.12 to the restriction of f to ∂ΩT (k) , and Z ΩT (k) hk(z) = f dµz , z ∈ ΩT (k) ∂ΩT (k)

TOME 66 (2016), FASCICULE 4 1720 A. Bendikov, L. Saloﬀ-Coste, M. Salvatori & W. Woess

defines a harmonic function on ΩT (k) which is continuous up to the boundary values of f. By Proposition 4.6, hk = f on every bifurcation line Lv with v ∈ V T (k). Now consider hk+1 : to this function, we can apply Lemma 4.3 on the set ε Ω (k) ⊂ Ω : there is a constant M(k) such that hk+1(z) M(k)hk+1(˜z) T T(k+1) 6 ˜ ˜ for every z ∈ ΩT(k), where z is the element on Lo with Re z = Re z. But on Lo , the function hk+1 coincides with f, which satisfies the growth condition (4.9) by Proposition 4.6. Therefore also hk+1 satisfies (3.8) on the whole of ΩT(k), with possibly modified constants. But now we can apply the uniqueness statement of Theorem 3.11, which yields that on ΩT(k), the function hk+1 is the Poisson integral of f taken over ∂ΩT(k) . In other words, the restriction of hk+1 to ΩT(k) is hk . If z ∈ HT then there is k such that z ∈ ΩT (k) , and setting h(z) = hk(z), we get a well-defined, positive ∆-harmonic function on HT that coincides with f on LT. Uniqueness is straightforward: if h˜ is any other positive ∆-harmonic function that extends f, then by Proposition 4.2 it satisfies the growth condition (3.8) on the whole of HT. We can apply once more the uniqueness ˜ statement of Theorem 3.11 on any ΩT (k) and get that h concides with hk = h on the latter set, for any k. The statement (ii) itself now follows a posteriori. Remark 4.8. — The results of this section also apply to the Laplacian H ∆α,β on H(q) = HT(q, 1), where the metric tree is R with vertex set Z. In S this case, LT is better denoted LH = k Lk ⊂ H, the union of all bifurcation lines in upper half plane. Note that Aff(H, q) acts simply transitively on LH, so that this group can be identified with LH. The random walk (Zτ(n)) is governed the the probability measure µe on L1 ∪ L−1 which is the exit distribution of (Zt) from the double strip S0 ∪ S1 when Z0 = i . α,β When we start with ∆HT on HT(q, p), then in view of Proposition 2.4 we H need to consider ∆α,βp here. In this situation, µe is the image of µ under the projection πH. What we have proved for µ on HT is also true for µe on H. Of course, we shall call the associated harmonic functions on LH ≡ Aff(H, q) the µe-harmonic functions.

5. Decomposition of positive harmonic functions

The aim of this section is to provide tools for giving a complete description of all positive ∆-harmonic functions on the whole of HT. For this

ANNALES DE L’INSTITUT FOURIER HARMONIC FUNCTIONS ON TREEBOLIC SPACE 1721 purpose, we shall need an understanding of the geometric boundary at infinity of HT. We quickly review the basic facts; a detailed description can be found in [3, Section 5]. The boundary (space of ends) ∂T was described in the lines after (2.7). The end compactification is Tb = T ∪ ∂T . The topology is such that a sequence (yn) converges to a point y in the metric tree T if it converges with respect to the original metric, while it converges to an end ξ ∈ ∂T if the geodesics o zn and o, ξ share initial pieces whose graph length tends to infinity. The boundary of H(q), which is just hyperbolic half plane metrically, is ∂H = R ∪ {∞}. The topology of Hb = H ∪ ∂H is the classical hyperbolic compactification; it is maybe better understood when one considers the Poincaré disk model instead of upper half plane, where Hb is the Euclidean closure (the closed unit disk), and ∞ corresponds to the north pole, while the remaining part of the unit circle corresponds to the lower boundary line R of upper half plane. Now HT(q, p) is a subspace of the direct product H(q) × Tp , so that the most natural geometric compactification HTc of HT is its closure in Hb × Tb. As mentioned, a detailed description of the resulting boundary ∂HT and of the different ways of convergence to the boundary can be found in [3].

Transience of (Xt) implies that the Green kernel Z ∞ (5.1) gα,β(w, z) = g(w, z) = hα,β(t, w, z) dt (w, z ∈ HT , w 6= z) 0 is strictly positive and ﬁnite, compare with [2]. The function w 7→ g(w, z) is harmonic on HT \{z}. Along with Proposition 4.2, the next uniform Harnack inequalities for the Green kernel are again a consequence of [2, Theorem 4.2].

Proposition 5.1. — With the same constants Cd as in Proposition 4.2 (in particular, Cd → 1 when d → 0) g(w, z0) g(z0, w) C and C , g(w, z) 6 d g(z, w) 6 d 0 0 whenever dHT(z, z ) 6 d and min{dHT(z, w), dHT(z , w)} > 10(d + 1). Positive harmonic functions can be described via the Martin boundary. The regularity properties of our Green kernel allow us to use the following approach, which is a special case of the potential theory (more precisely, Brelot theory) outlined in the book of Constantinescu and Cornea [11, Chapter 11]. See also Brelot [5], and in the Markov chain setting, compare with Revuz [24, Chapter 7].

TOME 66 (2016), FASCICULE 4 1722 A. Bendikov, L. Saloﬀ-Coste, M. Salvatori & W. Woess

We fix the “origin” o = (i , o) of HT as the reference point. A function + + f ∈ H is called minimal if f(o) = 1 and whenever f1 ∈ H is such that f > f1 then f1/f is constant. The minimal harmonic functions are the extremal points of the convex set {f ∈ HT+ : f(o) = 1}, which is a base of the convex cone HT+ of all positive harmonic functions. Every positive harmonic function is a convex combination (with respect to a Borel measure) of minimal ones. To make this more precise, define the Martin kernel by g(w, z) (5.2) k(w, z) = g(o, z) Then the Martin compactification is the unique (up to homeomorphism) minimal compactification of the underlying space (i.e., HT) such that all functions k(w, ·), w ∈ HT, extend continuously. The extended kernel is also denoted by k(·, ·). The Martin boundary M = M(∆) is the set of ideal boundary points in this compactification (i.e., the new points added “at infinity”). It is compact and metrizable. From the general theory [11], it is known that every minimal harmonic function arises as a boundary function k(·, ξ), where ξ ∈ M. The (Borel) set Mmin of all ξ ∈ M for which k(·, ξ) is minimal is called the minimal Martin boundary. The Poisson-Martin representation theorem says that for every h ∈ H+ there is a unique Borel measure νh on M such that ν(M\Mmin) = 0 and Z (5.3) h = k(·, ξ) dνh(ξ) . M If hH is an ∆H-harmonic function on H then by Proposition 2.4, the function hH ◦ πH is ∆-harmonic on HT, and the analogous property holds when we lift a ∆T-harmonic function hT from T to HT. Combining both cases, the function h(z) = hH(z) + hT(w) (where z = (z, w) ∈ HT) is also ∆-harmonic on HT. The first main purpose of this section and the machinery set up so far is to prove the following.

Theorem 5.2. — Every positive ∆α,β-harmonic h on HT has the form h(z) = hH(z) + hT(w) , z = (z, w) ∈ HT ,

H H T where h is non-negative ∆α,βp-harmonic on H and h is non-negative T ∆α,β-harmonic on T.

For the proof, we shall use the crucial fact that ∆α,β is invariant under the group A = A(q, p) deﬁned in Theorem 2.5, that is, tg(∆α,βf) = ∆α,β(tgf)

ANNALES DE L’INSTITUT FOURIER HARMONIC FUNCTIONS ON TREEBOLIC SPACE 1723

for all f ∈ Dom(∆α,β) and all g ∈ A, where

(5.4) tgf(z) = f(gz). Let

G(w, z) = g(w, z)φα,β(z) , where φα,β is as in (2.2). We have k(w, z) = G(w, z)/G(o, z), and it follows from (2.11) that G(gw, gz) = G(w, z) for all w, z ∈ HT (w 6= z) and g ∈ A . We shall use speciﬁc subgroups of A. First, in the tree, consider once more the k-th predecessor ok of the root vertex. Thus, ok ∈ H−k . Now deﬁne

B(T) = {g ψ = (idH, ψ): ψ ∈ Aff(T) , ψok = ok for some k ∈ N} . This is the embedding into A of the horocyclic subgroup of Aff(T), which consists of all isometries of the tree that leave each horocycle invariant. The latter is the analogue of the group of all “horizontal” translations in Aff(H, q), that is, of all mappings gb = (1, b), where b ∈ R, acting on H by z 7→ z + b. Thus, we define

B(H) = {g b = (gb, idT): b ∈ R} , which is of course isomorphic with the additive group R acting on H.

Theorem 5.3. — If h is a minimal ∆α,β-harmonic function on HT, then (at least) one of the following holds. H H (a) There is a minimal ∆α,βp-harmonic function h on H such that h(z, w) = hH(z) for all z = (z, w) ∈ HT, or T T (b) there is a minimal ∆α,β-harmonic function h on T such that h(z, w) = hT(w) for all z = (z, w) ∈ HT. Proof. — We know that h = k(·, ξ) for some ξ ∈ M, the Martin boundary of HT with respect to ∆α,β. There must be a sequence of points zn = (zn = xn + i yn, wn) ∈ HT such that dHT(o, zn) → ∞ and

h(z) = lim k(z, zn) for every z ∈ HT . n→∞

Using compactness, we may suppose that (zn) converges to a boundary point in HTc . Then we either have that wn → $ in Tb, or else yn is bounded below by a positive constant and zn → ∞ ∈ ∂H in the topology of Hb, that is, yn → +∞ or yn → y0 ∈ R and |xn| → +∞. 0 0 Case 1. wn → $. Let z = (z, w) and z = (z, w ) ∈ HT be two distinct 0 0 points with πH(z) = πH(z ) = z, so that h(w) = h(w ) in T. Consider the 0 conﬂuent v = w f w ∈ V (T). Since wn → $, we must have for all but

TOME 66 (2016), FASCICULE 4 1724 A. Bendikov, L. Saloﬀ-Coste, M. Salvatori & W. Woess

finitely many n that wn ∈ T\Tv , where Tv is the subtree Tv of T rooted at 0 0 v (i.e., consisting of all v ∈ T with v f v = v). There is an isometry of Tv that exchanges w and w0 (and fixes v). It extends to an element ψ in the horoyclic subgroup of Aff(T), that fixes every point in T\Tv. In particular, ψwn = wn for all but finitely many n. Therefore

0 0 G(z , zn) = G(g ψ z, g ψ zn) = G(z, zn) , whence k(z , zn) = k(z, zn) for all but ﬁnitely many n. Thus, as n → ∞, we ﬁnd h(z0) = h(z) for all z, z0 0 H with πH(z) = πH(z ). This means precisely that there is a function f on H such that h(z, w) = hH(z) for all z = (z, w) ∈ HT. It is now straightforward H H that h must be minimal ∆α,βp-harmonic.

Case 2. zn → ∞ ∈ ∂H and inf yn = y0 > 0. Let b ∈ R. By (2.8) and standard properties of the hyperbolic metric, dHT(g b zn, zn) = dH (b + xn) + i yn, xn + i yn

= dH(b + i yn, i yn) 6 dH(b + i y0, i y0) =: db .

Let Cdb be the corresponding Harnack constant in Proposition 5.1. Then, using that G(·, ·) is A-invariant and that φα,β(g b zn) = φα,β(zn)

G(g b z, zn) G(g b z, g b zn) k(g b z, zn) = G(g b z, g b zn) G(o, zn)

g(g b z, zn) = k(z, zn) 6 Cdb k(z, zn). g(g b z, g b zn) Letting n → ∞, we obtain

tbh(z) := h(g b z) 6 Cdb h(z) for all z ∈ HT .

Now, along with h, also tbh is positive harmonic, and minimality of h implies that the function tbh/h is constant. On the other hand, write z = (x + i y, w). If we let y → +∞ (and simultaneously h(w) = logq y → +∞) then d(g b z, z) = dH(b + i y, i y) → 0. Consequently, Proposition 4.2(a) implies that h(g b z)/f(z) → 1. Therefore tbf = f. This holds for every b ∈ R. As in Case 1, this is equivalent with the fact that there is a function hT on T such that h(z, w) = hT(w) for all z = (z, w) ∈ HT. It is again straightforward T T that h must be minimal ∆α,β-harmonic. Remark 5.4. — If both cases (a) and (b) of the last theorem occur R simultaneously, then this means that there is a minimal ∆α,βp-harmonic function hR on R such that h(z, w) = hR h(w) for all z = (z, w) ∈ HT. In the above proof, this happens when zn → (∞, $).

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(1) (2) Proof of Theorem 5.2.— Let Mmin and Mmin be the sets of all minimal harmonic functions on HT which are as in Theorem 5.3(a) and (b), respec- tiveley. Since the topology of the Martin boundary is the one of uniform convergence on compact sets, both are easily seen to be Borel sets in M. (1) (2) Also, Mmin \Mmin ⊂ Mmin. If f is any positive harmonic function on HT with integral representation (5.3), then we can set Z Z f H = k(·, ξ) dνh(ξ) . and f T = k(·, ξ) dνh(ξ) . (1) (1) Mmin M\Mmin Then f H(z, w) depends only on z, while f T(z, w) depends only on w, and f is their sum, as proposed. Remark 5.5. — In view of Theorem 4.7 and Remark 4.8, as well as Lemma 4.1 (which also applies to the metric tree R with vertex set Z), we have the following: a function on any of our spaces HT, H, T, or R is minimal harmonic for the respective one of our Laplacians if and only its restriction to LT, LH, V (T), or Z (respectively) is minimal harmonic for the respective induced random walk. As a ﬁrst application of Theorem 5.2, we can clarify when the weak Liouville property holds, that is, when all bounded harmonic functions are constant. For the following, we recall from [3, Thm. 5.1] that 1 log q a − 1 lim dHT(Xt,X0) = |`(α, β)| almost surely, where `(α, β) = , t→∞ t E(τ) a + 1 with τ and a given by (2.13) and (2.14), respectively. In particular, `(α, β) = 0 if and only if βp qα−1 = 1.

Theorem 5.6. — Suppose that p > 2. Then the Laplacian ∆α,β on HT has the weak Liouville property if and only if `(α, β) = 0. This follows from Theorem 5.2 in combination with the following.

Proposition 5.7. T (a) ∆α,β has the weak Liouville property on T if and only if `(α, β) 6 0. H (b) ∆α,βp has the weak Liouville property on H if and only if `(α, β) > 0. (Note the striking diﬀerence in the range of validity of the weak Liouville property on HT(q, p) with p > 2 as compared to H(q) = HT(q, 1) ! Also, compare those results with Karlsson and Ledrappier [23], [22].) Proof. — We note that the weak Liouville property is the same as triv- iality of the Poisson boundary, or equivalently, that the constant function 1 is minimal harmonic.

TOME 66 (2016), FASCICULE 4 1726 A. Bendikov, L. Saloﬀ-Coste, M. Salvatori & W. Woess

(a) It follows from [10] that the weak Liouville property holds for the random walk (Wτ(n)) on T if and only if `(α, β) 6 0. Via Lemma 4.1, this T transfers to ∆α,β . Statement (a) also follows from Proposition 5.8 below.

(b) Among the diﬀerent possible approaches, we adapt the method of Theorem 5.3. Since H(q) = HT(q, 1), everything that we have stated and proved for harmonic functions and the Green kernel on HT also applies to H. (The decomposition result for positive harmonic functions becomes of course trivial.) In particular, we have the Green kernel GH(w, z) = φ(α, βp)(z) gH(w, z), where φ(α, βp)(x + i y) = (βp)blogq yc yα. We also have the associated Martin kernel kH(w, z). A basic theorem in Martin boundary theory says that kH(·,Zt) converges almost surely to a minimal harmonic function. We now show that when `(α, β) > 0 then with positive probability, there is a random sequence (Zt(n)) such that kH(·,Zt(n)) → 1. Thus, the constant function 1 on H must be minimal harmonic. To show what we claimed, in the case when `(α, β) > 0 then the random walk (Yτ(n)) has positive drift, so that it tends to ∞ almost surely. When `(α, β) = 0, it is recurrent: with probability 1, it visits every point in Z. Thus it is unbounded and has a random subsequence that tends to ∞.

In both cases, we have a random sequence t(n) such that Im Zt(n) → ∞ almost surely.

Let us abbreviate zn = Zt(n) , so that kH(·, zn) tends to a minimal harmonic function h, and Im zn → ∞. We can use exactly the same method as in Case 2 of the proof of Theorem 5.3 and get that h(z + b) = h(z) for every z ∈ H and every b ∈ R. Thus, h only depends on Re z, which R means that it arises from lifting a minimal ∆α,βp-harmonic function eh to

H via h(z) = eh(logqIm z). We can apply Lemma 4.1 to the speciﬁc case R where the tree is just R, with vertex set Z : we see that the minimal ∆α,βp harmonic functions arise by interpolating the minimal harmonic functions for the random walk Yτ(n) on Z. The latter minimal harmonic functions are easily computed and are as follows. When `(α, β) = 0 then that random walk is recurrent, so that all its positive harmonic functions are constant. Thus, h = 1, as required. When `(α, β) > 0, the random walk has precisely two minimal harmonic functions, namely the constant function 1 and the function fb(k) = a−k (a well known exercise). We have to interpolate fb according to the variant of Lemma 4.1 where p = 1 and T = R with vertex set Z. This yields the R unique ∆α,pβ-harmonic extension bh. Now we have h(z) = bh(logqIm z).

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Thus, with probability one, kH(·,Zt) converges either to the constant 1 or to the latter function h. Suppose that it converges to h with probability 1. But then, the Poisson boundary has to consist of only one point, and in this case, every bounded harmonic function ought to be a multiple of h - a contradiction since h is unbounded. Thus, with positive probability, the limit is the function 1 which thus must be minimal harmonic on H. (A posterori, the limit function must be 1 almost surely.) Theorem 5.2 tells us that for describing all positive harmonic functions on HT, we need to know all such functions on T and on H. Here, “describing” means that we need to determine all minimal harmonic functions on each of those two spaces. This task is very easy on T. In view of Remark 5.5, we ﬁrst consider the induced random walk on V (T). The minimal harmonic functions for any transient nearest neighbour random walk on (the vertex set of) a tree are completely understood since the seminal article of Cartier [9]. For (Wτ(n)) on V (T), we have the following explicit formulas, taken from [25, p. 424–425].

Proposition 5.8. — The minimal Martin boundary of the random walk on V (T) coincides with the whole Martin boundary, which is ∂T. For the transition probabilities of (4.1), and with a as in (2.14), the Martin kernels on V (T) are given as follows, where we set b = max{a, 1} and c = min{a, 1/a}/p . For the boundary point $,

−h(v) kT(v, $) = b , v ∈ V (T) . For ξ ∈ ∂∗T, −h(v) h(o ξ)−h(v ξ) kT(v, ξ) = b c f f , v ∈ V (T) . The minimal harmonic functions on the metric tree T are the respective extensions of these functions according to (4.3), which are also denoted kT(w, ξ), where w ∈ T and ξ ∈ ∂T.

We see indeed (as already mentioned) that the constant function 1 on T is minimal harmonic if and only if a 6 1, that is `(α, β) 6 0. Describing the minimal harmonic functions on H is in general more com- plicated and less explicit. There is one exception which we describe next. H H Namely, if βp = 1 for the Laplacian on HT, then its projection ∆α,βp = ∆α is nothing but the ordinary hyperbolic Laplacian on upper half plane H with vertical drift parameter α, as in the second line of Deﬁnition 3.2. The

TOME 66 (2016), FASCICULE 4 1728 A. Bendikov, L. Saloﬀ-Coste, M. Salvatori & W. Woess bifurcation lines disappear in its analysis, and the minimal harmonic functions on H are well known; see e.g. Helgason [21], or many other sources.

H H Proposition 5.9. — The minimal Martin boundary of ∆α = ∆α,1 on H coincides with the whole Martin boundary, which is ∂H. The associated Martin kernels on H are given by the following extended Poisson kernels. For the boundary point ∞, max{1−α,0} Pα (x + i y), ∞ = y . For any boundary point ζ ∈ R, max{ α ,1− α } ζ2 + 1 2 2 P (x + i y), ζ = ymax{1−α,0} . α (ζ − x)2 + y2 At last, we can deduce the following result.

Theorem 5.10. — Consider HT(q, p) with p > 2 and the Laplacian α−1 ∆α,β with β = 1/p, so that a = q . (I) If α 6= 1 then the minimal harmonic functions on HT are parametrised by R ∪ ∂∗T and are given by ∗ z = (z, w) 7→ kT(w, ξ) , ξ ∈ ∂ T , and z = (z, w) 7→ Pα(z, ζ) , ζ ∈ R . (II) If α = 1 then the minimal harmonic functions are as in (I) plus, in addition, the constant function 1.

Proof. — We ﬁrst show that for ζ ∈ R, the function z = (z, w) 7→ Pα(z, ζ) is minimal harmonic. Suppose that Pα(z, ζ) > h(z, w) for all (z, w) = z ∈ HT, where h is non-negative harmonic on HT. We decompose h(z, w) = hH(z) + hT(w) according to Theorem 5.2. By minimality of Pα(·, ζ) on H, we have hH = c · Pα(·, ζ) for some c ∈ [0 , 1]. If c = 1 then we are done. If c < 1, 1 Z P (z, ζ) hT(w) = kT(w, ξ) dν(ξ) for all (z, w) ∈ HT , α > 1 − c ∂T where ν is a Borel measure on ∂T. We set w = o and z = x + i , where x ∈ R. Then Pα(x + i , ζ) > ν(∂T ) for all x. When x → ∞, the Poisson kernel tends to 0, so that the measure ν vanishes. Thus, hT ≡ 0 and h is a multiple of our Poisson kernel. Next, we need that for ξ ∈ ∂∗T , the function z = (z, w) 7→ kT(w, ξ) is minimal harmonic. This is completely analogous to the above, exchanging the roles of T and H. When α = 1, we know from Theorem 5.6 that the constant function 1 is also minimal harmonic. When α 6= 1, we still need to show that the

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two functions (z, w) 7→ kT(w, $) and (z, w) 7→ Pα(z, ∞) are not minimal harmonic. These are the functions 1 and (x+i y, w) 7→ y1−α. The first one is not minimal, because there are non-constant bounded harmonic functions. The second one is not minimal because when α > 1 it is already non- minimal on H, while when α < 1 is is already non-minimal on T. Outlook. — For a similar description of all minimal harmonic functions in the case when β¯ = βp 6= 1, one needs to study the boundary theory of “sliced” Laplacians ∆H(α, β¯). A natural approach is to apply The- orem 4.7 to that case. In view of Remark 4.8, this means that one wants to determine the minimal harmonic functions, or even the entire Martin compactification, for the random walk on Aff(H, q) with law µe. Since µe has super-exponential moments and continuous density with respect to Haar measure on that group, the guiding results are those of Elie [16], [17]. One expects that the minimal harmonic functions in the case `(α, β) 6 0 are given by an exponential depending only on the height (imaginary part) plus the Radon-Nikodym derivatives of translates of the unique µe-invariant measure on the real line. When `(α, β) > 0 one should first invert the drift by conjugating with a suitable exponential, and then apply the negative- drift case. However, at a closer look, one finds that Elie’s methods do not apply directly to our disconnected subgroup of the affine group. A second possible approach is to clarify how the methods of Ancona [1] apply to ∆H(α, β¯) in the “coercive” case β¯ qα−1 6= 1, while in the case β¯ qα−1 = 1 one might at- tempt to adapt the methods of Gouëzel [19], [20] concerning Martin boundary of hyperbolic groups to our non-discrete situation. Thus, additional work is required – this will be done separately.

Acknowledgement. — We thank Sara Broﬀerio for helpful interaction.

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[25] W. Woess, “Lamplighters, Diestel-Leader graphs, random walks, and harmonic functions”, Combin. Probab. Comput. 14 (2005), no. 3, p. 415-433. [26] ——— , “What is a horocyclic product, and how is it related to lamplighters?”, Internat. Math. Nachrichten of the Austrian Math. Soc. 224 (2013), p. 1-27, http: //arxiv.org/abs/1401.1976.

Manuscrit reçu le 24 mars 2015, révisé le 15 novembre 2015, accepté le 21 décembre 2015.

Alexander BENDIKOV Insitute of Mathematics Wroclaw University Pl. Grundwaldzki 2/4 50-384 Wroclaw, Poland [email protected] Laurent SALOFF-COSTE Department of Mathematics Cornell University Ithaca, NY 14853, USA [email protected] Maura SALVATORI Dipartimento di Matematica Università di Milano Via Saldini 50 20133 Milano, Italy [email protected] Wolfgang WOESS Institut für Diskrete Mathematik Technische Universität Graz Steyrergasse 30 A-8010 Graz, Austria [email protected]

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